forecasting with judgment and models

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Forecasting with Judgment and Models

Preliminary and incomplete.

First complete draft: May, 2006

This version: March 19, 2007.

Francesca Monti∗

ECARES, Universite Libre de Bruxelles

CP 114 Av.F.D. Roosevelt, 50.

B-1050 Bruxelles

[email protected]

Abstract

This paper proposes a parsimonious and model-consistent method for combiningforecasts generated by structural micro-founded models and judgmental fore-casts. The goal is to produce forecasts that are model-based, and thereforedisciplined by the rigor of the economic model, but that can also incorporate

judgmental information. In our set-up, there are three actors: the economicagents and two types of forecasters, the purely-model based and the judgmentalforecasters. They all know the true model of the economy, but their informationsets differ. The economic agents observe shocks as they realize and make theirdecisions consequently, while the forecasters do not observe current shocks. The

judgmental forecasters however have access to more timely information than thepurely model-based forecasters, but their forecasts are affected by some noise(i.e. they are not perfectly rational). Thus, the idea is to extract such informa-tion from the judgmental forecasts. This method also allows interpreting the

judgmental forecasts through the lens of the model. We illustrate the proposedmethodology with a real-time forecasting exercise, using a stripped-to-the-boneversion of an RBC model and the Survey of Professional Forecasters.

∗I am greatly indebted to Domenico Giannone, Lucrezia Reichlin and Philippe Weil forinvaluable guidance and advice. I am grateful to Gunter Coenen, David De Antonio Liedo,Marco Del Negro, Andrea Ferrero, Mark Gertler, Simon Potter, Paulo Santos Monteiro, ArgiaSbordone, Frank Schorfheide, Andrea Tambalotti and all seminar participants at the ECB,University of Pennsylvania, Federal Reserve Bank of NY for comments and useful discussions.The usual disclaimer applies.



JEL Classification: C32, C53Keywords: Forecasting, Judgment, Kalman filter, Real time

1 Introduction

Much of the macroeconometric literature of the last decade has focused onmaking micro-founded dynamic stochastic general equilibrium (DSGE) modelsa viable option for policy analysis and forecasting. Since Smets and Wouters(2004) have shown that DSGE models estimated with Bayesian techniques seemto perform quite well in forecasting relative to standard benchmark models suchas VARs, DSGE models are playing a more relevant role in practice and haveindeed become an increasingly important tool for policy analysis and forecastingat central banks. The attractiveness of using these models to forecast derivesfrom the fact that they are theoretically consistent and based on first princi-ples. The micro-foundations imply that the parameters are more likely to betruly structural and allow interpreting the forecasts in an economically intuitiveway. Moreover, the structural nature of the model allows computing forecasts

conditional on a policy path and allows examining the structural sources of theforecast errors and their implications for monetary policy.

Despite their growing employment in practice, model-based forecasts stillseem to be outperformed at short horizons -and particularly in the nowcast1-by forecasts produced by institutional and professional forecasters, such as theFederal Reserve’s Greenbook (e.g. Sims, 2003) or the Survey of ProfessionalForecasters.2 Where does this advantage come from?

Professional forecasters monitor and analyze literally hundreds of data se-ries, using informal methods to distill information from the available data. Notonly they access what is generally called hard data (data series that are re-leased by the statistical agencies, as for example, GDP, industrial production,etc), they also gather so-called soft information, i.e. things like the quantityof goods transported by railway in each month (Bruno and Lupi, 2004) or the

electricity consumed each month (Marchetti and Parigi, 1998). Moreover, pro-fessional forecasters are able to incorporate new data and new information asit becomes available throughout the month or the quarter and therefore arecan take advantage of the timeliness of this information. Indeed, as Giannone,Reichlin, Small (2005) point out, timely information seems to play a very im-portant role in improving the quality of the forecasts, and of the nowcasts inparticular. Finally, in their forecasts, professional forecasters account also forall sheerly judgmental information. A typical example is the adjustments of theforecasts made in 1999 in order to account for the fear of the Y2K bug. Indeedthis seemed at the time a very important event, but since it had never happenedno model could be expected to encompass it, while the institutional forecasterscould.

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Nowcasts are estimates of the current value of variables, such as GDP, that are unknownin the current period due to information lags2This view has recently been challenged by Edge, Kiley and Laforte (2006), who suggest

that a richly specified DSGE models have a forecasting performance that is comparable tothat of the Greenbooks. We believe that their results are very much related to the samplethey choose for their out of sample exercise, i.e. 1996-2001. We will discuss this in more detailin the empirical application.

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Hence, judgement - i.e. information, knowledge and views outside the scopeof a particular model 3 - strongly informs the institutional forecasts. The empir-ical evidence at hand suggests that the ability to account for more, more timelyand ”softer” information is what makes the professional or judgmental (I willuse the two terms interchangeably from now on) forecasts better at nowcastingand forecasting short horizons.

The introduction of DSGE models in a policy and projection environmenthas given rise to a literature on how the model’s outcomes should be combinedwith judgmental input and off-model information. The aim of this paper is topropose a method for combining judgment - proxied by judgmental forecasts -with model-based forecasts, in order to make predictions that are more accuratebut nevertheless disciplined by rigorous economic theory. In particular, wepropose to interpret the judgmental forecasts as an estimate - made with adifferent, possibly more informative, information set - of the real signal, estimatewhich can be filtered in order to extract the information it possibly contains.We then use the model to generate another forecast that can now also accountfor judgmental information and therefore make more accurate predictions. Thenew forecast that we generate is a combination of the model-based forecast andthe judgmental forecast: the Kalman filter will automatically associate weightsto the two forecasts depending on the information content of the judgmentalforecasts. Moreover, with the methodology we propose, we will be able tolook at the judgmental forecasts through the lens of the model. Storytelling isdifficult when it comes to judgmental forecasts; in our set-up we will be able tointerpret the forecasts in light of the model and therefore somehow structuralizethe forecasts. The approach we propose is similar in spirit to the one used byCoenen, Levin and Wieland (2005) and Koenig (2005) to deal with revisions.One of the key factors that distinguishes our approach from theirs is that weconsider the judgmental forecasts as optimal forecasts made with a differentinformation set, not a noisy signal of the actual variables.

Recently, other authors have addressed the issue of how to use soft dataand judgment in models. Svensson (2005), Svensson and Tetlow (2005) and

Svensson and Williams (2005) develop different frameworks that allow account-ing for central-bank judgment when constructing optimal policy projections of the target variables and the instrument rate. They show that such monetarypolicy may perform substantially better than monetary policy that disregards

judgment and follows a given instrument rule. Our approach differs quite sub-stantially from theirs: our goal is solely to produce model-based forecasts thatcan account for judgmental and off-model information. Our approach leavesthe structure of the DSGE model unchanged and combines the model-basedforecasts with the judgmental forecasts.

In a Bayesian framework, Robertson, Tallman and Whiteman (2005) suggesta minimum relative entropy procedure for imposing moment restrictions on sim-ulated forecasts distributions from a variety of models. This technique involveschanging the initial predictive distribution to a new one that satisfies specific

moment conditions that come from outside of the models, i.e. that are judg-mental. Therefore, minimum-entropy methods allow adjusting the full posteriordistribution of the DSGE models to match a given experts’ assessment.

Another approach that can be used to incorporate judgmental and off-model

3This definition appears in Svensson (2005)

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information is that of Boivin and Giannoni (2005). They build on the factormodel literature, started by Stock and Watson (1999, 2002) and Forni, Hallin,Lippi and Reichlin (2000), and propose an empirical framework for the estima-tion of DSGE models that exploits the relevant information from a data-richenvironment. Their methodology allows using as much information as possi-ble to estimate the structural model and to update the estimates of the state

variables featuring in the model. In this way, soft information can be used sys-tematically to update the current assessment of the state variables as well as of the short-term forecast.

The paper is structured as follows. In Section 2 we outline the frameworkand describe the proposed methodology; we illustrate how to extract the weightsgiven to the model-based and the judgmental forecast; and describe how tostructuralize the professional forecasts. In Section 3 we apply the proposedmethodology on a stripped-to-the-bone version of an RBC model using theSurvey of Professional Forecasters’ forecasts to extract eventual judgmental in-formation. Section 4 presents the results of the empirical application describedin the previous section. In Section 5 we give some conclusions and outline futureextensions of this paper.

2 The Econometric Methodology

2.1 The Framework

Let us consider a general linear(ized) rational expectations model of the form

AE t

zt+1Z t+1

= B

ztZ t

+ Cxt (1)

xt = M xt−1 + εt (2)

εt ∼ W N (0, Q)

where zt is a vector of non predetermined endogenous variables, Z t is a vector of

predetermined endogenous variables or of lagged exogenous variables such thatE tZ t+1 = Z t+1, xt is a vector of exogenous variables following the process (2), Qis diagonal and A, B, C and M are conformable matrices of coefficients that forma structural parameter space that we shall call Θ. Linear(ized) general equi-librium models containing additional lags, lagged expectations or expectationsfarther in the future can be cast in this form simply by expanding the vectorszt and Z t appropriately. Several numerical techniques have been developed tosolve models of the form (1)-(2), see, e.g., Blanchard and Kahn (1980), Klein(1997) and Sims (2000). The solution of the model has the following state-spacerepresentation

S t =

Z txt

= F (θ)S t−1 + H (θ)εt (3)

zt = N (θ)S t, (4)

where F (θ), H (θ) and N (θ) are functions of the underlying structural parame-ters.

To allow for greater generality we augment each equation in (4) with a (pos-sibly serially correlated) residual, or error term, as in Ireland (2004). The model

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now consists of the transition equation (3) and the new observation equation

yt = zt + vt = N (θ)S t + vt, (5)

wherevt+1 = Dvt + ξt+1 (6)

for all t = 1, 2,... and ξt is a vector of zero mean, serially uncorrelated inno-vations that is normally distributed with covariance matrix Eξtξ′t = V and isuncorrelated with the innovation εt. There are two appealing features in thisset-up. First, depending on the way in which the matrices D and V are defined,the residuals can be interpreted as measurement errors or as additional elementscapturing all of the movements and co-movements in the data that the DSGEmodels, because of their elegance and parsimony, cannot explain. Second, themodel consisting of (3),(5) and (6) overcomes the well-known stochastic singu-larity problem, pointed out by Ingram et al.(1994), that often appears in DSGEmodels, deriving from the assumption that few fundamental shocks drive all thedynamics of the model.

Model (3),(5) and (6) can be rewritten more compactly as:

st+1 =

S t+1vt+1

= G(θ)st + ν t+1 (7)

yt = Λ(θ)st (8)

where G(θ) =

F (θ) 0

0 D

, Λ(θ) =

N (θ) I

and ν t =

H (θ)εt

ξt

is seri-

ally uncorrelated, normally distributed with zero mean and covariance matrix

Eν tν ′t = Q =

H (θ)ΣH (θ)′ 0

0 V

.

From now on, for notational simplicity, we will drop the indication that thematrices G, Λ, etc... are function of the structural parameters θ.

Associated with the state space representation (7)-(8) is the innovations

representation

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st|t = Gst−1|t−1 + Kut (9)

yt = ΛGst−1|t−1 + ut (10)

where st|t = E [st|yt, yt−1,...,y0, s0] is the estimate of the state vector st basedon the observations of yτ up to date t , ut = yt − yt|t−1 = yt − E [yt|yt−1,...,y0]the forecast error made when forecasting yt given the observations of yτ up todate t-1, K = GP Λ′(ΛP Λ′)−1 is the steady state Kalman gain, and P is uniquepositive semidefinite solution that satisfies the algebraic Riccati equation

P = Q + GP G′ − GP Λ′(ΛP Λ′)−1ΛP G′. (11)

P is the steady-state covariance matrix of the innovations st − st|t−1 given theinformation in period t −1. It is useful to rewrite the innovations representation

4The conditions for the existence of this representation are stated carefully, among otherplaces, in Anderson, Hansen, McGrattan, and Sargent (1996). The conditions are that that(F,H,N) be such that iterations on the Riccati equation for Σt = E (xt−xt)(xt−xt)′ converge,which makes the associated Kalman gain K t converge to K. Sufficient conditions are that(F, N ′) is stabilizable and that (F ′,H ′) is detectable. See Anderson, Hansen, McGrattan,and Sargent (1996, page 175) for definitions of stabilizable and detectable.

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given by (9) and (10) as the sum of two components: one that is forecastablegiven the information set containing all observations of yτ up to date t-1,

M t = Sp{yt−1, yt−2, ...y0}, (12)

and a sequence of innovation terms. That is,

st+h|t+h = Gh+1st−1|t−1 +

h−1j=0

GjKut+h−j (13)

yt+h = ΛGh+1st−1|t−1 + ΛG

hi=1

GiKut+h−i + ut+h. (14)

2.2 Model of the Professional Forecasts

The goal of this section is to show how to incorporate judgmental forecastsinto models of the form (7)-(8), or equivalently in (9)-(10). In order to do so,we need to somehow formalize these forecasts. More specifically, we need tomake assumptions on the model and the information set that the professional

forecasters use to generate their forecasters. In this paper, we will make theassumption that the professional forecasters know the structure of the economy,i.e. know the model (7)-(8) and can use it to forecast, but they access aninformation set, I t, which is more informative than M t.

The assumptions on the information available in each period t are outlined inTable 1. We assume that the shocks hit the economy at the beginning of periodt. The agents observe the shocks and base their decisions on this knowledge. Wethen assume that there are three types of forecasters. The first type generateshis forecasts on the basis of the model and the data released by the statisticalagency. The data reporting agency releases data about the current period at theend of the period, so in period t there is data available only up to t-1. Thereforethe information set available to the first type of forecaster in time t comprisesexclusively information up to time t-1: his information set is M t, as defined in(12). From now on I will call the first type of forecaster ’purely’ model-basedforecaster.

The second type of forecaster also knows the model of the economy anduses it to make its forecasts, but accesses another information set I t, whichcomprises M t but is possibly more informative. This type represents the pro-fessional forecasters (PF from now on). As highlighted in the Introduction,their information set is plausibly richer than M t: they collect soft, intra-periodextra-model information, such as monthly electricity consumption or quantityof goods transported by railway in each month. In what follows, we assumethat professional forecasters have extra-model information only on the currentperiod’s shock but not on future shocks. In appendix we illustrate the case inwhich we assume that they have some extra-model information, not exclusively

on the current shocks, but possibly also on future shocks. Considering the waythe professional forecasts are made and the type of extra-information they use -i.e., information about this month electricity consumption, but also informationon future tax raises or on the possible effects of extraordinary events like theY2K bug or the World Cup- , both cases are credible. However, the possible in-consistency arising from assuming that the Professional Forecasters know more

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Table 1: Information structure

t t + 1

shocks hit the economy

agents observe them

PF collect information

and release their forecast

stat. agency releases

data on period t

about the future that the agents, but that the agents don’t incorporate them intheir solution persuaded me to relegate this part to the appendix.

Finally, the third type of forecaster will use the method I propose: I willdefine the forecasts they produce augmented forecasts, since the use the PF toaugment their information set.

Let us formalize rigorously the second type of forecaster, the professionalforecasters. PF are able to obtain information on the current period’s shocks.At any given time T their information set I T comprises M T but is such that,for h = 1, 2, 3, 4

E [uT I ′T ] = 0, (15)

uT +h ⊥ I T

The forecasters know the model of the economy and produce their forecastsas linear least squares forecasts given their information set plus an error term.5

Notice that if we just modelled the PF’s forecasts as a noisy version of the actualsignal, i.e.

st+1+h = Gst+h + ν t+1+h

yPF t+h = Λst+h + et+h

this formulation would be totally inconsistent with our assumptions. First of all,the et+h’s would not be forecast errors - since they are not orthogonal to the past- and therefore the PF would not be forecasts, but rather noisy signals of actualfuture variables. This would mean that we are assuming that the PF have crystalballs through which they see the future. This is neither realistic, nor model-consistent. Instead we want to model the output of the professional forecastersas forecasts, as shown in detail below. Sargent (1989) first distinguished amongthese two possible modelizations, discussing two models of a statistical agencythat is collecting and reporting observations on a dynamical linear stochasticeconomy.

For t = 1, 2,...,T − 1 both purely model-based forecasters and professionalforecasters are going to construct the innovations representation (9)- (10). Fort T , professional forecasters will report the following: for h=0,

sT |T = GsT −1|T −1 + KuT (16)

yPF t = E [yT |I T ] + ηT |T (17)

where E [yT |I T ] = ΛGsT −1|T −1 + E [uT |I T ] is the least squares forecast madeby the PF with their information set I t and ηT |T is the measurement error (the

5For a recent review of the debate on the rationality (unbiasedness and efficiency) of macroeconomic forecasts see Schuh(2001)

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typo) made by the professional forecasters in T while reporting their forecast.This can be cast in state-space form as follows. For h=0,

sT |T = GsT −1|T −1 + KuT (18)

yPF t = ΛGsT −1|T −1 + wT |T ,

wherewT |T = E [uT |I T ] + ηT |T ,

and KuT wT |T

∼ W N

0 , Σ0

with

Σ0 =

KE (uT u

′T )K ′ KE (uT w

′T |T )

E (wT |T u′T )K ′ EwT |T w

′T |T

.

For h = 1, 2, 3, 4 we have

sT +h|T +h = GsT +h−1|T +h−1 + KuT +h

yPF T +h = ΛGsT −1|T −1 + ΛGhKE [uT |I T ] + ηT +h|T , (19)= ΛGsT +h−1|T +h−1 + wT +h|T

where

wT +h|T = −Λh−1j=1

GjKuT +h−j − ΛGhK (uT − E [uT |I T ]) + ηT +h|T (20)

KuT +hwT +h|T

∼ W N

0 , Σh

and

Σh = KE (uT +hu′T +h)K ′ 0

0 EwT +h|T w′T +h|T

.

As above, ηT +h|T is the measurement error (the typo) made by the professionalforecasters in T while reporting their forecast for period T + h. We assume thatηs|T ⊥ uτ , for any s and τ , and that ηT +h|T ⊥ E (uT |I T ) for h = 0, 1, 2, 3, 4.Clearly, the form of the matrices Σh depends crucially on the assumptions wemade on the information set of the professional forecasters. Here Σh will beblock diagonal for all h = 0, since we have modeled the PF as having someextra-information only on the current shock. In appendix we present the casein which the PF can have extra-information up to 4 periods ahead. In thatcase, the Σh will not necessarily be block diagonal: the value of the off-diagonalterms will depend on the information on future shocks actually carried by theprofessional forecasts.

We want to extract the optimal linear projection of yT +h given I T , i.eE [yT +h|I T ], from yPF T +h. To understand how the augmented forecasts are built,consider the fact that models (18) and (19) can be seen as a new state spacemodel in which the new observables are the professional forecasts. By filteringmodel (18)-(19) with a time-varying Kalman smoother one can obtain optimal

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estimates of the state variables s+T +h|I T

that comprise the extra-information con-

tained in the professional forecasts and employ it optimally within the model.6

Having s+T +h|I T

for we can finally construct the augmented forecasts y+T +h|I T

y+T +h|I T

= Λs+T +h|I T

(21)

that incorporates optimally in the model-based framework the judgemental in-formation coming from the conjunctural forecasters.

In order to implement the Kalman filter, we need to be able to recover theexact form of the covariance matrices Σ0 and Σh for h=1,2,3,4. To recover allthe elements of Σ0, we proceed as follows. First of all let us point out that,since ηT |T ⊥ uT by assumption, then

E (uT w′T |T ) = E [uT E (uT |I T )

′] + E [uT η′T |T ] = E [uT E (uT |I T )

′].

The element on the right hand side of this equation can be calculated as follows.First notice that the following equality holds

E [uT yPF ′T ] = E [uT (ΛGsT −1|T −1)′] + E [uT E (uT |I T )

′] + E [uT η′T |T ]

= E [uT E (uT |I t)′],

The second equality derives from the fact that uT ⊥ ΛGsT −1|T −1 by construc-tion and that ηT ⊥ uT by assumption. Finally, as the series for the uT ’s arereadily available via the Kalman filter, we are able to recover empirically thevalue of E [uT y

PF ′T ], and therefore of E [uT E (uT |I T )

′]. Moreover, notice that,since E (uT |I T ) is a linear projection of uT on Sp(I T ), the space spanned by I T ,then

uT = E (uT |I T ) + µT

where µT is orthogonal to the space spanned by I T . Therefore,

E [uT E (uT |I T )′] = E [E (uT |I T )E (uT |I T )

′], (22)

i.e. we have determined also the variance of the expected value of the current

shock given the information set I T , E (uT |I T ), and we showed it is equal to thecovariance among the shock and its expected value.

In order to recover

E [wT |T w′T |T ] = E [E (uT |I T )E (uT |I T )

′] + E [ηT |T η′T |T ], (23)

(the equality holds because ηT |T ⊥ E (uT |I T ) by assumption), we will first definethe forecasters’ forecast error as

eT = yT − yPF T

= uT − wT |T .

= uT − E (uT |I T ) − ηT |T .

Its variance, whose value can be recovered from sample data, is:

E (eT e′T ) = E (uT u

′T ) − E [uT E (uT |I T )

′] + (24)

− E [uT E (uT |I T )′]′ + E [wT |T w

′T |T ]

6The notation s+T +h|IT

mean the estimate of the state sT +h made in T give the augmented

information set I T .

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E (uT u′T ) can be obtained by the Kalman filter on the system of equations (7)

and (8) as followsE (uT u

′T ) = ΛP Λ′

Where P is the solution of the Riccati equation defined in (11). Using the aboveequations, we can finally recover E [wT |T w

′T |T ] and we therefore have pinned

down all the values of the matrix Σ0.Moreover, it is possible to recover the value of the variance of the typoE [ηT |T η

′T |T ]; from (23) and (24) we infer the following equation, which can be

reshuffled to obtain E [ηT |T η′T |T ].

E (eT e′T ) = E (uT u

′T ) − E [E (uT |I T )E (uT |I T )

′] + EηT |T η′T |T . (25)

The procedure to recover Σh, for h=1,2,3,4 is very similar.

2.3 Model-consistent weights for forecast pooling

Now we discuss how the time-varying Kalman smoother we use to generatethe ”judgment-augmented” forecasts actually combines the judgmental fore-

casts with the purely model-based forecasts. Let us turn back to the system of equations we smooth to obtain the augmented forecasts, i.e.

sT +h|T +h = GsT +h−1|T +h−1 + KuT +h (26)

yPF T +h = ΛGsT +h−1|T +h−1 + wT +h|T ,

where for h=0wT |T = E [uT |I T ] + ηT |T (27)

and for h=1,2,3,4,

wT +h|T = −Λh−1j=1

GjKuT +h−j − ΛGhK (uT − E [uT |I T ]) + ηT +h|T

and KuT +hwT +h|T

∼ W N

0 ,

Σh11 Σh

12

Σh21 Σh

22

.

In period T we filter (26) and we generate a new innovations representation,initialized with s+

T −1|I T = sT −1|T −1

s+T +h|I T

= Gs+T +h−1|I T

+ K 2haT +h (28)

yPF T +h = ΛGsT +h−1|I T + aT +h,

where

s+T +h|I T

= E (s+T +h|I T

|yPF T +h, yPF T +h−1,...,yPF T , yT −1, ....y1, sT −1|T −1),

aT +h = yPF T +h − E

yPF T +h|yPF T +h, yPF T +h−1,...,yPF T , yT −1, ....y1, sT −1|T −1

.

The Kalman gain K 2h is time-varying and takes the form: for h=0,

K 20 = Σ012Σ0

22−1

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and

K 2h = (GP h|h−1G′Λ′ + Σh12)(ΛGP h|h−1G′Λ′ + Σh

22)−1 (29)

otherwise. In order to understand how the Kalman filter is combining the purelymodel-based forecast and the judgemental forecast, let us consider the case of the nowcast:

s+T |I T

= (I − K 20)GsT −1|T −1 + K 20yPF T , (30)

where s+T −1|I T

= sT −1|T −1 and GsT −1|T −1 is the purely model-based forecast of

the state at time T. Therefore the augmented nowcast for yT is

y+T |I T

= Λs+T |I T

= Λ(I − K 20)GsT −1|T −1 + ΛK 20yPF T +h (31)

Since K 20 has the form described in equation (29), the weight given to the judgmental forecast yPF T is directly proportional to Σ0

12, i.e. to the correlationamong uT and wT |T , and inversely proportional to the variance of wT |T Σ0

22.That is, the more the professional forecasters are able to gather information onthe current period’s shock, the more the Kalman filter will use the professionalforecasts when combining the two forecasts, but it will down-weigh them if thevariance of their forecast errors is too large. Similarly for higher horizons.

The assumption that the professional forecasters have information only onthe current period shocks is crucial in determining the negligible weights as-signed to the professional forecasts for h = 0. In appendix we present someresults for the case in which the PF are assumed to have some off-model infor-mation on current and future shocks up to four periods ahead. In that case, theweight associated to the judgmental forecast will be sizeable -depending on theinformational content, of course- at all horizons considered.

2.4 Using the model to interpret judgemental forecasts

Another interesting aspect of this procedure is that it also allows to see the judgmental forecasts through the lens of the model. Storytelling is difficult whenit comes to judgmental forecasts; in our set-up we will be able to interpret theforecasts in light of the model and therefore somehow structuralize the forecasts.

Let us have another look at model (28). The element K 20aT , with

aT = yPF T − E

yPF T |s+T −1|I T

,

is the estimate of the current period’s structural shocks made by the professionalforecasters with their information set I T . That is, these are the structural shocksthe professional forecasters perceive given their information set; they do notnecessarily coincide with the ”real” structural shocks. Indeed

K 20aT = E [KuT |I T ].

Moreover, given this information, we can also derive [uT |I T ] as follows:

E [uT |I T ] = (K ′K )−1K ′E [KuT |I T ]

= (K ′K )−1K ′(K 20aT ).

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Now, recalling that the professional’s forecasts are represented by equations(18)-(19), we can evaluate exactly how much of the forecast is due to extrainformation on the current shocks and how much of it is instead due to mea-surement errors. Moreover, we can construct different scenarios - e.g. assumethat the professional forecasters have extra information only on certain typesof shocks but not on others - and compare them among each other. As we will

show in Section 4, this can be very informative.

3 An application

In order to illustrate the methodology proposed in the previous section, we willapply it to a stripped-to-the-bone version of an RBC model with unit root tech-nology. Let us briefly outline the model. A representative consumer has pref-erences defined over consumption C t during each period t=1,2,..., as describedby the expected utility function

E

∞

i=1β t ln(C t) (32)

where the discount factor satisfies 0 < β < 1. In this economy there is only onefinal good Y t and it is produced using capital K t and labor N t according to theconstant-returns-to-scale technology

Y t = (AtN t)αK

(1−α)t , (33)

where 0 < α < 1 and At is a labor-augmenting technological change process. Wewill assume that labor is supplied inelastically and that there is no populationgrowth: in such case N t is constant for any t and we can normalize it to 1, i.e.

N t = 1 for any t . We there for rewrite equation (33) as

Y t = Aαt K

(1−α)t . (34)

The logs of the technology shock At follow a first order autoregressive processof the form:

ln(At) = ln(At−1) + γ + εt, (35)

where the innovation εt is serially uncorrelated and normally distributed withmean zero and standard deviation σ.

In each period the representative agent decides how much of output Y t toconsume and how much to invest, subject to the resource constraint

Y t = C t + I t. (36)

By investing I t units of output in period t, the agent increased the capital stockK t+1 available in period t + 1 according to

K t+1 = (1 − δ)K t + I t, (37)

where δ is the depreciation rate and satisfies 0 < δ < 1.The standard method of analyzing models with steady state growth is to

transform the economy into a stationary one where the dynamics are more

11



tractable. The transformation, which is shown in great detail in King, Plosserand Rebelo (1988a, 1988b), involves dividing all variables in the system by thegrowth component, which in our setting corresponds to At.The stationarizedmodel is very similar to the untransformed model, with some exceptions thatwe will highlight in what follows.

Let us start defining E [At+1

At] = γ A, yt = Y t

At, ct = C t

At, kt = Kt

Atand so on.

The stationary version of the model defined by equations (32), (34), (35), (36)and (37) is

max E

∞i=1

β t ln(C t) (38)

subject to

yt = k(1−α)t , (39)

yt = ct + it, (40)

and(γ Aeεt+1)kt+1 = kt + it. (41)

The first order conditions for this problem are

Rt = [(1 − α)k−αt + (1 − δ)] (42)

where Rt is the gross rate of return of capital in t and

c−1t = βE t[c−1t+1Rt+1

eεt+1]. (43)

Equation (43) equates the marginal rate of substitution to the marginal productof capital for all t=1,2,..... Equations (39)-(43) form a system of five non-linearstochastic difference equations in the model’s five variables yt, ct, kt, it, rt. Alinear approximation of this system can be derived log-linearizing it around itssteady state, as shown in Appendix A. The system of equations (39)-(43) hasan approximate solution of the form:

xt+1 =

kt+1kt

= G(θ)xt +

10

εt+1 (44)

zt =

∆yt

ct − ytit − yt

= Λ(θ)xt, (45)

where the expressions for the matrices G(θ) and Λ(θ) in terms of the structuralparameters θ can be found in Appendix A. A particular feature of this modelis that one shock - the aggregate technology shock εt - drives all business cyclefluctuations.

Any equation of the form (45) can be rewritten in terms of differences, simplyby premultiplying everything with a suitable matrix:

yt =

∆yt∆ct∆it

=

1 0 01 1 − L 01 0 1 − L

zt =

1 0 01 1 − L 01 0 1 − L

Λxt. (46)

In general, equation (44) will change conformably to take into account that (46)is now defined in terms of current and lagged states st. In this specific case the

12



state equation will not need to be rewritten, since it already contains the laggedstate.

We also augment each equation in (46) with a serially correlated residual,or error term, so that the model now consists of (44),

yt = ∆yt

∆ct∆it

= Λ

∗

xt + vt, (47)

where Λ∗ =

1 0 0

1 1 − L 01 0 1 − L

Λ and

vt+1 = Dvt + ξt+1 (48)

for all t = 1, 2,... and ξt is a vector of zero mean, serially uncorrelated inno-vations that is normally distributed with covariance matrix Eξtξ′t = V and isuncorrelated with the innovation εt.

We have considered three possible specifications of vt, the first one assumesvt is white noise, the second one allows for autocorrelation (but not serial cor-relation) in vt, while in the third specification we allow vt to be a VAR(1). Theresults were very similar for each of the three specifications, therefore we willpresent results only for the model with autocorrelated but not cross-correlatedresiduals, which is our best performing one. All results are robust to the speci-fication of the measurement error vt.

We calibrated all the model’s parameters, with the exclusion of the varianceof the technological shock and and the parameters describing the measurementerrors, using values from King, Plosser and Rebelo (1988a, 1988b) and Ireland(2004). All parameters are reported in Table 7 in Appendix A. We estimate thevariance of technological shock and the parameters of the measurement erroronly once, using maximum likelihood, and then we take them as calibrated. Thecovariance matrices Σi in (20) are instead estimated using a rolling window.

For the estimation and the forecasting exercise we use real-time quarterlydata for real GDP and real consumption for the US from the Philadelphia Fedreal-time dataset. The dataset covers the period 1947 through 2005 and thefirst available vintage is 1965:Q4. Due to the unavailability of real-time dataon population, we have made the somewhat heroic assumption that the popu-lation has been constant throughout the period considered. In what follows, wewill perform an out-of-sample real-time forecasting exercise, using as evaluationsample the period 1987-2004.

We use the Survey of Professional Forecasters (SPF) as example of judgmen-tal forecast. The Survey of Professional Forecasters, conducted by the FederalReserve Bank of Philadelphia, is based on many individual commercial forecasts,which are then grouped in mean or median forecasts. The Survey is conductednear the end of the second month of each quarter and publishes forecasts for

the current quarter and the next 4 quarters in the future. We consider a sampleperiod going from the second quarter of 1987 to the fourth quarter of 2004.

An important data-related issue regards the appropriate ”actual” series touse when comparing the various forecasts. Because macroeconomics data iscontinuously revised, we need to make a choice about which revision to use.Following Romer and Romer (2000), we choose to use the second revision, i.e.

13



Table 2: Relative MSFE of forecasts of GDP growth with respect to naivebenchmark. Asterisks denote forecasts that are statistically more accurate thanhe naive benchmark at 1% (∗ ∗ ∗), 5%(∗∗) and 10%(∗)

relative to constant growthEVALUATION SAMPLE: 1987:2 - 2004:4

Forecast horizon RBC SPFQ0 0.86 0.68 ∗∗Q1 0.88 ∗∗ 0.77Q2 0.92 0.88Q3 0.94 ∗∗ 0.94Q4 0.97 0.95


Forecast horizon RBC SPFQ0 0.81∗∗ 0.80∗∗Q1 0.85 ∗ ∗ ∗ 0.94Q2 0.84∗ ∗ ∗ 1.01

Q3 0.84 ∗ ∗ ∗ 1.01Q4 0.87 ∗∗ 1.01

the one done at the end of the subsequent quarter. The second revision seems tobe the appropriate series to use because it is based on relatively complete data,but it is still roughly contemporaneous with the forecasts we are analyzing. Thisseries does not include rebenchmarking and definitional changes that occur inthe annual and quinquennial revisions and should, therefore, be conceptuallysimilar to the series being forecast.

Let us now present some forecasting results that will highlight the motivationof this paper. We will compare all the forecasts of GDP and Consumptiongrowth to a naive benchmark: the constant growth model, i.e. random walk inlevels. Tables 2 and 3 report the performance in out-of-sample forecasting of the forecasts generated with the RBC model, specified as having autocorrelated,but not serially correlated residuals (i.e. D is diagonal) and of the SPF relativeto the naive benchmark, for GDP growth and consumption growth respectively.Each table is divided in two subtables which consider the full sample period1987:Q2-2004:Q4 and the subsample 1996:Q2-2004:Q4. In the first column of each table one can see the ratio of the mean square error of the purely model-based forecast (RBC) against the mean square error of the naive benchmark,while the second column reports the ratio of the mean square error of the SPFagainst the mean square error of the naive benchmark. Asterisks indicate arejection of the test of equal predictive accuracy between each forecast and thenaive benchmark. 7

7Following Romer and Romer (2000), our inference is based on the regression: (zht−zmht)2−

(zht− znaiveht

)2 = c+uht where z is the variable to be forecasted at horizon h using model -m .The estimate of c is simply the difference between forecast-m and a Naive model MSFEs, andthe standard error is corrected for heteroskedasticity and serial correlation over h-1 months.This testing procedure falls in the Diebold-Mariano-West framework, and Giacomini andWhite (2006, Section 3.2, see in particular Comment 4) show that by using rolling window

14



Table 3: Relative MSFE of forecasts of Consumption growth with respect tonaive benchmark. Asterisks denote forecasts that are statistically more accuratethan he naive benchmark at 1% (∗ ∗ ∗), 5%(∗∗) and 10%(∗)


Forecast horizon RBC SPFQ0 0.97 0.70Q1 0.92 ∗∗ 0.89Q2 0.94 ∗∗ 0.94Q3 0.94 ∗ 0.99Q4 0.94 ∗ 1.00


Forecast horizon RBC SPFQ0 0.89 ∗∗ 0.96Q1 0.89 ∗∗ 1.06Q2 0.88 ∗∗ 1.03

Q3 0.87 ∗∗ 1.03Q4 0.86 ∗∗ 1.08

Over the full sample period 1987:Q2-2004:Q4, SPF forecasts of GDP growthoutperform the model-based forecasts at all horizons, but their advantage re-duces as the forecasting horizon grows. Considering the subsample 1996:Q2-2004:Q4, we notice the model seems to perform better at all horizons excludingthe nowcast. The model-based forecasts of consumption growth are poorer inthe very short run than in the medium term (4 quarters ahead). Over the fullsample period 1987:Q2-2004:Q4, SPF forecasts of consumption growth outper-form the model-based forecasts in the nowcast and 1 period ahead, and neverin the subsample 1996:Q2-2004:Q4. The two results on consumption growthpoint to the added-value of enforcing accounting identities, especially in themedium-term.

Figure 1 reports the smoothed forecast errors for the nowcast of GDP (cen-tered moving average 4 quarters on each side) over the full sample period. Modeland judgement seem to contain information that is useful in different points intime: in some periods the model does better than the benchmark and in oth-ers it does worse, and the same holds for SPF. Particularly during recessions,

judgement seems to fare much better than the model (a similar result can befound in Giannone, Reichlin and Sala, 2005).

In the last part of the sample, and particularly between 1996 and 2000, judgmental forecasts have performed quite bad, while the model seems to fareacceptably. A similar result can be found in Edge, Kiley and Laforte (2006),

where the authors compare the forecasting performance of a richly specifiedDSGE models to the Greenbooks in the sample 1996-2000 and find that themodel’s forecasting performance is comparable to the one of the Greenbooks.

estimators, as we do here, the limiting behavior of this type of tests is standard, and thereforestandard asymptotic theory can be used for inference on the difference in predictive accuracy.

15



Figure 1: NOWCASTS: Smoothed Square forecast errors

Q1−90 Q3−92 Q1−95 Q3−97 Q1−00 Q3−02

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

NOWCAST: Smoothed Square Forecast errors (relative to constant growth)

SPF

RBC

AUG

In fact, not only the richly specified DSGE model that Edge, Kiley and Laforte(2006) propose, but also the toy model I use seem to fare better than judgmentin that period.8

Furthermore, Figure 1 highlights how performances of the purely model-based and the judgmental forecasts do not seem positively correlated, but ratherthey seem to somehow counterbalance each other. Therefore, it is plausible that

combining the two forecasts can be advantageous. The method I propose indeedallows for a model-based and hence interpretable averaging of the two forecasts.

In the following section we present the results obtained when applying themethodology proposed in Section 2 to the toy model presented above and usingthe SPF forecasts.

4 Results of the Forecasting Exercise

The main goal of this section is to present model-based forecasts for real GDPand real consumption that can account for the judgmental information containedin the SPF forecasts. We will compare their performance on the basis of their

mean square forecast error, i.e. deeming better a forecast with a smaller MSE.We will perform out-of-sample forecasting exercises on the full evaluation sample1987:Q2-2004:Q4 and on the subsample 1996:Q2-2004:Q4.

8It is worth stressing that the naive benchmark model Edge, Kiley and Laforte use forGDP growth, i.e. a random walk, is not model compatible.

16



Table 5: Weight given to the SPF forecast of ∆GDP and ∆CONS in theaugmented forecast . 2004:4

∆GDP ∆CONS

nowcast 0.3941 0.2574

1 step ahead -0.1038 -0.0020

2 step ahead -0.0341 0.0018

3 step ahead -0.0185 0.0019

4 step ahead -0.0078 0.0014

in the nowcast. The reason is that the weights given by the Kalman filter tothe SPF tend to zero as their information content goes to zero. Finally, noticethat the weight associated to the SPF forecast of consumption growth is smaller

than those associated to the SPF forecast of GDP growth. This indicates thatthe extra-model information accessed by the professional forecasters is moreinformative on GDP than on consumption.

Let us briefly compare these results with the ones reported in appendix forthe case in which the SPF are assumed to have extra some extra-information upto 4 periods ahead. In the latter case, the augmented forecasts beat both theSPF forecasts and the model-based forecasts at all horizons over the full eval-uation sample 1987:Q2-2004:Q4. Moreover, in the full sample, the augmentedforecasts built under the assumption that the PF have some extra-informationon the current and future shocks perform much better that the augmented fore-cast obtained by assuming that the PF have extra-model information only onthe current shock. Interestingly, the opposite is true in the subsample 1996:Q2-2004:Q4. This indicates that the SPF are better modeled by assuming that they

have extra-model information on current and future shocks in the full evaluationsample, while in the subsample 1996:Q2-2004:Q4 they are better represented byassuming they have extra-information only on the current shock. The above re-sults point out to the decline in predictability highlight in D’Agostino, Giannoneand Surico (2006).

Table 6 reports the mean square forecast error (MSE) of the purely model-based forecasts, the SPF and the augmented forecasts relative to the naive model(random walk in levels), when forecasting consumption growth.

Figures (2)-(8) report the nowcasts and the forecasts, plotted against thedata. In each figure, the green/grey solid line represents the actual data; thesolid line with circles is the series of the SPF forecasts, the dotted line portraysthe purely model-based forecasts; the dashed line corresponds to the augmentedforecasts. The latter forecasts becomes more and more similar to the model-

based forecast as we increase the forecasting horizon. This is not surprising,since we have assumed that the SPF have extra information only regarding thecurrent period; thus the informational value of their forecast is much lower athorizons higher than the nowcast and will therefore be given less weight. Thiswould clearly not be the case under different assumptions for the information

18



Q1−90 Q1−95 Q1−00

−0.5

0

0.5

1

1.5

Figure 2: Nowcast for GDP - EVALUATION SAMPLE 1987:2-2004:4.The green/grey solid line represents the actual data; the solid line with circles is theseries of the SPF forecasts, the dotted line portrays the fully model-based forecasts;the dashed line corresponds to the the augmented forecasts

from the technological shocks are the shocks to the residual term that is meantto describe all the dynamics that is not captured by the RBC model. Howeverthe proposed procedure would, when applied to a more elaborate DSGE model,allow for understanding the perception of the different shock - monetary, fiscal,etc.. - that the SPF have.

Finally, interesting counterfactual exercises can be done in this setup. Figure11 reports, for the period 1987:2-1994:4, the the actual SPF nowcast and thenowcasts they would have done if they had only parts of the information theyactually have. In particular, the black solid line represents actual real GDPgrowth; the starred line is the actual SPF nowcast. The line marked withplus signs is the nowcast the SPF would have made if they would have had noinformation on any of the shocks, the line with squares plots the nowcast theywould have made if they had extra-information only on the technological shock,while the line with diamonds is the nowcast they would have made if they hadextra-information only on the measurement error shocks.

From this figure we can extract some very relevant information. For exam-ple, consider the fall in actual GDP growth in the last quarter of 1989. Theactual SPF nowcast for that quarter is very close to the actual figure and, in-

terestingly, coincides with the nowcast the SPF would have made if they onlysaw the technology shock (green line). On the other hand, the nowcast the SPFwould have made if they only saw the ”measurement error” shock (red line)coincides with the one they would have made if the had no extra-informationat all (blue line). This means that this specific movement in GDP growth wasmost probably due to a technological shock. Similarly, in the third quarter of

20



Q1−88 Q1−90 Q1−92 Q1−94 Q1−96

−0.5

0

0.5

1

1.5

Figure 3: Nowcast for GDP - PERIOD I 1987:2-1996:1. The green/greysolid line represents the actual data; the solid line with circles is the series of the SPFforecasts, the dotted line portrays the fully model-based forecasts; the dashed linecorresponds to the the augmented forecasts

Q1−98 Q1−00 Q1−02 Q1−04

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Figure 4: Nowcast for GDP - PERIOD II 1996:2-2004:4. The green/greysolid line represents the actual data; the solid line with circles is the series of the SPFforecasts, the dotted line portrays the fully model-based forecasts; the dashed linecorresponds to the the augmented forecasts

21



Q1−90 Q3−92 Q1−95 Q3−97 Q1−00 Q3−02

−0.5

0

0.5

1

1.5

Figure 5: Forecasts 1 step ahead for GDP - EVALUATION SAMPLE

1987:2-2004:4. The green/grey solid line represents the actual data; the solid linewith circles is the series of the SPF forecasts, the dotted line portrays the fully model-based forecasts; the dashed line corresponds to the the augmented forecasts

Q1−90 Q3−92 Q1−95 Q3−97 Q1−00 Q3−02

−0.5

0

0.5

1

1.5

Figure 6: Forecasts 2 step ahead for GDP - EVALUATION SAMPLE1987:2-2004:4. The green/grey solid line represents the actual data; the solid linewith circles is the series of the SPF forecasts, the dotted line portrays the fully model-based forecasts; the dashed line corresponds to the the augmented forecasts

22



Q1−90 Q1−95 Q1−00 Q1−05

−0.5

0

0.5

1

1.5

Figure 7: Forecasts 3 step ahead for GDP - EVALUATION SAMPLE1987:2-2004:4. TThe green/grey solid line represents the actual data; the solidline with circles is the series of the SPF forecasts, the dotted line portrays the fullymodel-based forecasts; the dashed line corresponds to the the augmented forecasts

Q1−90 Q1−95 Q1−00 Q1−05

−0.5

0

0.5

1

1.5

Figure 8: Forecasts 4 step ahead for GDP - EVALUATION SAMPLE1987:2-2004:4. The green/grey solid line represents the actual data; the solid linewith circles is the series of the SPF forecasts, the dotted line portrays the fully model-based forecasts; the dashed line corresponds to the the augmented forecasts

23



Q1−90 Q1−95 Q1−00

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Nowcasts and confidence bands (no parameter uncertainty) 0 quarters ahead

Figure 9: Nowcasts with confidence bands (no parameter uncertainty)1987:2-2004:4. The black solid line is the purely model-based nowcast, while theblack dashed-dotted lines are its confidences bands. Similarly, the blue lines representthe augmented nowcast (and its confidence bands). When constructing the confidencebands, we only consider the uncertainty in the estimation of the state, but no param-eter uncertainty. the current period’s shock.

24



Q1−90 Q1−95 Q1−00

−0.5

0

0.5

1

1.5

techonology

measurement error on Dy

measurement error on Dc

Dy

Figure 10: Current structural shocks perceived by the SPF when now-casting 1987:2-2004:4. The dotted line represents actual real GDP growth. Theblue line is the technological shock the SPF see, the green line is the measurement

error they perceive on GDP growth, while the red line is the measurement error theyperceive on real consumption growth.

1992, the SPF nowcast coincides with the nowcast the SPF would have made if they only saw the ”measurement error” shock (red line), while have informationon the technological shock is like having no information. In this case, then,the variation of the GDP was certainly not due to technology, but due to the”measurement errors” and some other factors. Of course, this exercise wouldbe much more interesting if we developed it in a setup in which we were ableto distinguish, for example, monetary or fiscal shocks, but it is any how quiteinformative.

5 Conclusions and Extensions

In this paper we have proposed a method to incorporate judgmental informa-tion, proxied by professional forecasts, into model-based forecasts. We suggestedmodeling the professional forecasts as optimal estimates of the variables of inter-est, made with a different, possibly more informative, information set; we thenhave shown how they can be accounted for in the framework of a linearized andsolved DSGE model. The methodology we propose allows generating forecaststhat are more accurate than the purely model-based ones, but that are stilldisciplined by the economic rigor of the model.

We have also highlighted how to infer the information content of the SPF

forecasts from the weights that the Kalman filter assigns to them. In particular,the weights given by the Kalman filter to the SPF go to zero as their informationcontent goes to zero. More precisely, the more the professional forecasters areable to gather information on the shocks, the more the Kalman filter will usethe professional forecasts when combining them with the predictions from the

25



Q1−88 Q1−90 Q1−92 Q1−94

−0.5

0

0.5

1

1.5

Figure 11: Counterfactual exercise -nowcast- 1987:2-1994:4. The blacksolid line represents actual real GDP growth; the dashed-dotted line is the actual SPFnowcast. The blue line is the nowcast the SPF would have made if the would havehad no information on any of the shocks, the green line plots the nowcast they wouldhave made if they had extra-information only on the technological shock, while thered line is the nowcast they would have made if they had extra-information only onthe measurement error shocks.

26



model, but it will down-weigh them if the variance of their forecast errors is toolarge.

Finally we have described how to interpret the forecasts through the lens of the model. We were able to extract the structural shocks as they were perceivedby the professional forecasters and to make several counterfactual exercises onthe forecasts the professional forecasters would have done if they saw only some

of the shocks.We working on several extensions to this paper. First, in a joint paper with

Domenico Giannone and Lucrezia Reichlin we allow for the timely informationto enter the model directly, not processed by the professional forecasters. Wediffer from Boivin and Giannoni (2005) in that we consider a large datasetcontaining intra-period data, in order to really capture the effects of timelyinformation.

Then, we are also working on reformulating the problem so to be able toaccount for the possibility that the forecasts feedback into the model. If forexample we wanted to include a policymaker that targeted current inflationand current output gap, the estimate of inflation and output made using judg-mental information should feedback into the model via the policy rule. Thisextension can be implemented using the extension of the Kalman filter pro-posed by Svensson and Woodford (2003) that allows to do signal-extractionwith forward-looking obsverables10.

Once we have reformulated the problem, we will be able to apply the method-ology we propose to richer DSGE models, with more shocks. This can be veryinteresting from a storytelling perspective, because, as we have sketched in thesimple application we consider in this paper, it will allows us to understandwhich types of shocks the professional forecasters perceived. More importantly,we will be able to understand, in the cases in which the professional forecastersforecasts where very close to the actual data, the extent to which the perceptionof certain shocks helped them forecasting so well and infer if the movements inthe data were due to, say, a technology or a monetary shock.

10Since the forecasts feedback into the model, we cannot solve the model first, as we did inour simple RBC case, and then forecast. For this reason we will need to deal with forwarding-looking observables.

27



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6 Appendix A: Solving the RBC model

The equilibrium conditions for the optimization problem are

yt = k(1−α)t , (49)

yt =

ct +

it,

(50)(γ Aeεt)kt+1 = kt + it. (51)

Rt = [(1 − α)k−αt + (1 − δ)] (52)

c−ηt =

β

γ ηA

E t[c−ηt+1Rt+1

eεt+1]. (53)

In absence of shocks, the economy converges to the following deterministic bal-anced growth path/steady state that can be obtained from equations (49)-(53)dropping the time subscripts and through some simple manipulation.

R =γ ηA

β

Y

K =

γηA

β+ 1 − δ

1 − α

I

K = γ A − (1 − δ)

C

Y = 1 −

(1 − α)(γ A − (1 − δ))

R − (1 − δ)

I

Y =

(1 − α)(γ A − (1 − δ))

R − (1 − δ)

A linear approximation of system (49)-(53) can be derived log-linearizing itaround its steady state. Let us define, for and variable x, xt = ln(xt

X), i.e. the

log-deviation of xt from its steady state X. With some manipulation we obtainedthe following linearization for the system (49)-(53).

yt =C

Y ct +

I

Y it

kt+1 =I

γ AK it +

1 − δ

γ Akt − εt

yt = (1 − α)yt

rt =1 − α

R

Y

K (yt − kt)

0 = E t[−η(ct+1 − ct) + rt+1]

Manipulating it a bit, this system can also be rewritten all as a function of kt+1,ct and εt

+1.

kt+1 = λ1kt + λ2ct − εt (54)

E t[∆ct] =λ3

ηE tkt+1 (55)

yt = (1 − α)kt (56)

31



where

λ1 =R

γ A

λ2 = 1−R + α(1 − δ)

γ A(1 − α)

λ3 = − α(R − (1 − δ))R

Various methods are available for solving linear difference models like (54)-(56) under rational expectations, but given the simplicity of our model we donot need to resort to elaborate methods, we can simply apply the method of undetermined coefficients as described, e.g., in Campbell (1994). That is, we

”guess” the functional form of kt+1 and ct and then we verify it by findingparameters that satisfy the restrictions of the approximate loglinear model.

As pointed out in King, Plosser and Rebelo (1988b), if technology is a loga-rithmic random walk with drift, then the solution to the transformed economyis particularly simple. The only impact of technological progress is to reset thetransformed economy’s capital stock relative to its long-sum stationary level. A

positive 1% technological innovation in the untransformed economy leads to a1% decline in the transformed economy’s capital stock. Therefore we assumethat

kt+1 = µkt − εt+1 (57)

ct = πckkt (58)

and similarly, yt = πyk kt, etcetera. Substituting (57) and (58) into (54)-(56)and manipulating, we obtain a second order equation for πck which has onlyone positive solution (as required by the problem). Therefore we obtain

kt+1 = µkt − εt+1 (59)

ytct

=

πck(1 − α)

kt,

where

πck =−(λ1 − 1 + 1

η)λ2λ3 +

(λ1 − 1 + 1

η)λ2λ3)2 + 4λ2

λ1λ3η

2λ2

µ = λ1 + λ2πck.

In order to be able to bring the model to the data we still need to work onit a bit more. If we consider the variables in levels, we can write

ln yt = at + ln(Y ) + yt

ln ct = at + ln(c) + ctln it = at + ln(I ) + it

Since we are in fact looking at a balanced growth path rather than a steadystate, we cannot recover the steady state values Y, C, I, we can only pin down

32



some ratios as C Y

and Y K

. Therefore, in order to relate the model (59) to thedata, we rather look at

∆(ln yt) = ln γ A + (α + µ − 1)kt−1 − αkt

ln ct − ln yt = log(C

Y ) + [πck − (1 − α)]kt

ln it − ln yt = log( I Y

) + [πik − (1 − α)]kt. (60)

Finally we obtainkt+1

kt

=

µ 01 0

kt

kt−1

+

−10

εt+1 (61)

∆ ln yt

ln ct − ln ytln it − ln yt

=

ln γ A

log(C Y

)log( I

Y )

+

−α α + µ − 1

πck − (1 − α) 0πik − (1 − α) 0

kt

kt−1

,

in the form of a state-space econometric model, allowing the procedure for eval-uating the likelihood function to continue using the Kalman filtering algorithms

outlined, for example, by Hamilton (1994,Chapter 13).Table 7 reports the parameters values for the model with the autocorre-

lated but not cross-correlated residual. All other specifications have the samecalibrated parameters, but differ slightly in the estimated parameters.

Table 7: Parameter values for the model with the autocorrelated but not cross-correlated residual.

Parameter value

β 0.988 calibratedα 0.667 calibratedδ 0.01 calibratedη 1 (log utility) calibrated

γ 1.0072 calibratedσ2ε 0.01 estimated

Dyy 0.6692 estimatedDcc 0.2215 estimatedV yy 0.01 estimatedV cc 0.0075 estimatedV yc -0.0483 estimated

7 Appendix B: alternative modeling of the pro-

fessional forecasters

Here I show how to construct an augmented forecast that extracts and accountsfor the information contained in professional forecasts under the assumption thatthe professional forecasters have extra-model information on current and future

33



shocks.11 Their information set I T ⊇ M T and is such that, for h = 0, 1, 2, 3, 4

E [uT +hI ′T +h] = 0. (62)

In this case, the forecasters will report the following state-space form. Forh=0,

sT |T = GsT −1|T −1 + KuT (63)

E [yT |I T ] = ΛGsT −1|T −1 + wT |T ,

wherewT |T = E [uT |I T ] + ηT |T

and KuT wT |T

∼ W N

0 , Σ0

where

Σ0 =

KE (uT u

′T )K ′ KE (uT w

′T |T )

E (wT |T u′T )K ′ EwT |T w

′T |T

,

ηT |T is, as before, the measurement error (the typo) made by the forecastersin T while reporting their forecast for period T . The professional nowcast co-incides with the one generated under the assumption that the forecasters haveinformation only on the current period; the forecasts will instead be different.For h = 1, 2, 3, 4 we have

sT +h|T +h = GsT +h−1|T +h−1 + KuT +h

E [yT +h|I T ] = ΛGsT −1|T −1 +hi=1

ΛGiKE [uT +h−i|I T ] + ηT +h|T , (64)

= ΛGsT +h−1|T +h−1 + wT +h|T

where

wT +h|T = Λhi=1

GiK [E (uT +h−i|I T ) − uT +h−i] + E [uT +h|I T ] + ηT +h|T (65)

KuT +hwT +h|T

∼ W N

0 , Σh

and

Σh =

KE (uT +hu′T +h)K ′ KE (uT +hw′

T +h|T )

(wT +h|T u′T +h)K ′ EwT +h|T w

′T +h|T

.

As above, ηT +h|T is, as before, the measurement error (the typo) made by theforecasters in T while reporting their forecast for period T + h. We assume

11We acknowledge, but for now ignore the potential inconsistency arising from the fact of assuming that professional forecasters have information on future shocks, while agents do notand fail to look at the professional forecasters to have more information.

34



that ηs|T ⊥ uτ , for any s and τ , and that ηT +h|T ⊥ E (uT +i|I T ) for any i, h.Moreover we will make the following assumptions:

ut ⊥ E (uτ |I T ) ∀τ = t

E (ut|I T ) ⊥ E (uτ |I T ) ∀τ = t (66)

ut ⊥ E (uτ |I T ) ∀τ = t

which will allow us to recover Σ0 and Σh for h=1,2,3,4.Once we have recovered the exact form of the covariance matrices Σ0 and

Σh we can then smooth (63) and (64) with a time-varying Kalman smoother,in order to obtain optimal estimates s+

T +h|I T ,for h=0,1,2,3,4, that comprise the

extra-information contained in the forecasts and employs it optimally withinthe model. Therefore, using s+

T +h|I twe can create a forecast y+

T +h|I T ,

y+T +h|I T

= Λs+T +h|I T

, (67)

that incorporates optimally in the model-based framework the judgemental in-formation coming from the conjunctural forecasters.

Since the nowcasts coincides with the ones of the previous subsection, Σ0

can be recovered exactly in the same way. To recover all the elements of Σ h, weproceed as follows. Given the assumptions we made in (66), it is easy to showthat

E (uT +hw′T +h|T ) = E [uT +hE (uT +h|I T )

′].

On the basis of assumptions (66), the following equality holds

E [uT +hE (yT +h|I T )′] = E [uT +hE (uT +h|I T )′],

As the series for the uT +h are readily available via the Kalman filter, we areable to recover empirically the value of E [uT +hE (yT +h|I T )

′], and therefore of E [uT +hE (uT +h|I T )

′]. Moreover, notice that, since E (uT +h|I T ) is a linear projection of uT +h on the space spanned by I T , Sp(I T ), then

uT +h = E (uT +h|I T ) + µT

where µT is orthogonal to the space spanned by I T . Therefore,

E [uT +hE (uT +h|I T )′] = E [E (uT +h|I T )E (uT +h|I T )

′], (68)

i.e. we have determined also the variance of the expected value of the currentshock given the information set I T , E (uT +h|I T ), and we showed it is equal tothe covariance among the shock and its expected value.

In order to recover EwT +h|T +hw′T +h|T +h we will first define the forecasters’

forecast error as

eT +h = yT +h − E (yT +h|I T )

= uT +h − wT +h|T .

The variance of eT +h, whose value can be recovered from sample data, is:

E (eT +he′T +h) = E (uT +hu′T +h) − E [uT +hw′T +h|T ] +

− E [uT +hwT +h|T ]′ + E [wT +h|T w

′T +h|T ]

35



Table 9: Weight given to the SPF forecast of ∆GDP and ∆CONS in theaugmented forecast . 2004:4

∆GDP ∆CONS

nowcast 0.3941 0.2574

1 step ahead 0.3606 0.1261

2 step ahead 0.3236 0.0931

3 step ahead 0.3387 0.0832

4 step ahead 0.3321 0.0818

37

forecasting with judgment and models

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